The goal of decentralized optimization over a network is to optimize
a global objective formed by a sum of local (possibly nonsmooth)
convex functions using only local computation and communication. It
arises in various application domains, including distributed
tracking and localization, multi-agent co-ordination, estimation in
sensor networks, and large-scale optimization in machine
learning. We develop and analyze distributed algorithms based on
dual averaging of subgradients, and we provide sharp bounds on their
convergence rates as a function of the network size and
topology. Our method of analysis allows for a clear separation
between the convergence of the optimization algorithm itself and the
effects of communication constraints arising from the network
structure. In particular, we show that the number of iterations
required by our algorithm scales inversely in the spectral gap of
the network. The sharpness of this prediction is confirmed both by
theoretical lower bounds and simulations for various networks. Our
approach includes both the cases of deterministic optimization and
communication, as well as problems with stochastic optimization
and/or communication.